An asymptotic property of quaternary additive codes (Q6632038)

An additive code over a finite field is a code that is closed under addition but possibly not under scalar multiplication. Let \(n_k(s)\) be the maximal length \(n\) such that an additive code over \({\mathbb F}_q\) of length \(n\) and minimum distance \(n-s\) exists, where \(k= \ell/2\) and the code can be viewed as an \(\ell\) dimensional vector space over \({\mathbb F}_2\). The authors view an additive \([n,k,d]_4\) code as a multiset of \(n\) lines in \(PG(2k-1,2)\) such that each hyperplane contains at most \(s=n-d\) of the codelines. Let \(\lambda_k\) be the \(\lim \sup n_k(s)/s\) as \(s\) goes to infinity. The authors prove that \(\lambda_k = \frac{4^k-1}{4^{k-1}-1}\) for \(k \geq 1.5\) and determine when \(n_k(s) = \lambda_k\). They produce a new family of constant-weight additive codes with parameters \([4^k-1,k,4^k-4^{k-1}]_4 \) where \(k = \ell/2\) and \(\ell\) is an odd integer.